Quinary (base-5 or pental) is a

numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The same sequence of symbo ...

with

five 5 is a number, numeral, and glyph. 5, five or number 5 may also refer to: * AD 5, the fifth year of the AD era * 5 BC, the fifth year before the AD era Literature * ''5'' (visual novel), a 2008 visual novel by Ram * ''5'' (comics), an awa ...

as the base. A possible origination of a quinary system is that there are five digits on either hand. In the quinary place system, five numerals, from 0 to 4, are used to represent any

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

. According to this method,

is written as 10,

twenty-five Twenty-Five or 25 may refer to: *25 (number), the natural number following 24 and preceding 26 * one of the years 25 BC, AD 25, 1925, 2025 Music * ''Twenty Five'' (album), a 2006 greatest hits album by George Michael ** ''Twenty Five'' (DVD), rel ...

is written as 100 and sixty is written as 220. As five is a prime number, only the reciprocals of the powers of five terminate, although its location between two

highly composite number __FORCETOC__ A highly composite number is a positive integer with more divisors than any smaller positive integer has. The related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller ...

s ( 4 and 6) guarantees that many recurring fractions have relatively short periods. Today, the main usage of base 5 is as a

biquinary Bi-quinary coded decimal is a numeral encoding scheme used in many abacuses and in some early computers, including the Colossus. The term ''bi-quinary'' indicates that the code comprises both a two-state (''bi'') and a five-state (''quin''ary) ...

system, which is decimal using five as a sub-base. Another example of a sub-base system, is

sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...

, base 60, which used 10 as a sub-base. Each quinary digit can hold log₂5 (approx. 2.32) bits of information.

Comparison to other radices

Usage

Many languagesHarald Hammarström, Rarities in Numeral Systems: "Bases 5, 10, and 20 are omnipresent." use quinary number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot,

Luiseño The Luiseño or Payómkawichum are an indigenous people of California who, at the time of the first contacts with the Spanish in the 16th century, inhabited the coastal area of southern California, ranging from the present-day southern part of ...

and

Saraveca Saraveca is an extinct Arawakan language once spoken in Bolivia by the Sarave. It is saidEncyclopædia Britannica, "Numerals and numeral systems". to be the only language with a numeral system based exclusively on five, although quinary systems e ...

. Gumatj has been reported to be a true "5–25" language, in which 25 is the higher group of 5. The Gumatj numerals are shown below: However, Harald Hammarström reports "one would not usually use exact numbers for counting this high in this language and there is a certain likelihood that the system was extended this high only at the time of elicitation with one single speaker", pointing to the Biwat language as a similar case (previously attested as 5-20, but with one speaker recorded as making an innovation to turn it 5-25).

Biquinary

A decimal system with 2 and 5 as a sub-bases is called

, and is found in

Wolof Wolof or Wollof may refer to: * Wolof people, an ethnic group found in Senegal, Gambia, and Mauritania * Wolof language, a language spoken in Senegal, Gambia, and Mauritania * The Wolof or Jolof Empire, a medieval West African successor of the Mal ...

and Khmer.

Roman numeral Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, ea ...

s are an early biquinary system. The numbers 1, 5, 10, and 50 are written as I, V, X, and L respectively. Seven is VII and seventy is LXX. The full list is: Note that these are not positional number systems. In theory a number such as 73 could be written as IIIXXL without ambiguity as well as LXXIII and it is still not possible to extend it beyond thousands. There is also no sign for zero. But with the introduction of inversions such as IV and IX, it was necessary to keep the order from most to least significant. Many versions of the

abacus The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool which has been used since ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the Hi ...

, such as the

suanpan The suanpan (), also spelled suan pan or souanpan) is an abacus of Chinese origin first described in a 190 CE book of the Eastern Han Dynasty, namely ''Supplementary Notes on the Art of Figures'' written by Xu Yue. However, the exact design ...

and

soroban The is an abacus developed in Japan. It is derived from the ancient Chinese suanpan, imported to Japan in the 14th century. Like the suanpan, the soroban is still used today, despite the proliferation of practical and affordable pocket electr ...

, use a biquinary system to simulate a decimal system for ease of calculation. Urnfield culture numerals and some

tally mark Tally marks, also called hash marks, are a unary numeral system ( arguably). They are a form of numeral used for counting. They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate r ...

systems are also biquinary. Units of

currencies A currency, "in circulation", from la, currens, -entis, literally meaning "running" or "traversing" is a standardization of money in any form, in use or circulation as a medium of exchange, for example banknotes and coins. A more general def ...

are commonly partially or wholly biquinary. Bi-quinary coded decimal is a variant of biquinary that was used on a number of early computers including Colossus and the IBM 650 to represent decimal numbers.

Quadquinary

A vigesimal system with 4 and 5 as a sub-bases is found in Nahuatl.

Calculators and programming languages

Few calculators support calculations in the quinary system, except for some

Sharp Sharp or SHARP may refer to: Acronyms * SHARP (helmet ratings) (Safety Helmet Assessment and Rating Programme), a British motorcycle helmet safety rating scheme * Self Help Addiction Recovery Program, a charitable organisation founded in 19 ...

models (including some of the EL-500W and EL-500X series, where it is named the ''pental system'') since about 2005, as well as the open-source scientific calculator

WP 34S The HP 30b (NW238AA, variously codenamed "Big Euro", "Mid Euro" and "Fox") is a programmable financial calculator from HP which was released on 7 January 2010. The HP 30b is an advanced version of the HP's prior model HP 20b. F ...

. Python's int() function supports conversion of numeral systems from any base to base 10. Thus, the quinary number 101 is evaluated using int('101',5) as 26.

References

External links

Quinary Base Conversion
includes fractional part, from Math Is Fun * {{commons category inline, Quinary numeral system

uses

D'ni ''Myst'' is a franchise centered on a series of adventure video games. The first game in the series, ''Myst'', was released in 1993 by brothers Rand and Robyn Miller and their video game company Cyan, Inc. The first sequel to ''Myst'', ''Riven' ...

numerals from the

Myst ''Myst'' is a graphic adventure/puzzle video game designed by the Miller brothers, Robyn and Rand. It was developed by Cyan, Inc., published by Broderbund, and initially released for the Macintosh in 1993. In the game, the player's charact ...

franchise, integers only, fan-made. Positional numeral systems 5 (number)